Lame parameters

Note that both representations (13.96) and (13.97) hold true for a medium with variable Lame parameters (Aki and Richards, 2002). However, in a general case one should use the corresponding Green s tensors for the Lam6 equation with variable coefficients. 

To be able to represent arbitrary shapes, the orthogonal curvilinear coordinates Si and S2 on the middle surface are introduced together with the associated Lame parameters 2li and Sla, see for example Dym [69] or Novozhilov [134], The undeformed middle surface is characterized by the respective principal radii R and i 2 in agreement with the following condition ... 

The curvilinear coordinate s will be given such that it represents the arc length and therefore, just as in the case of the Lame parameters for the cylindrical shell in Section 6.2.2, the following is required ... 

Besides the Lame parameters, any pair of two of the following moduli can be used for a description of the elastic properties of an isotropic material ... 

Figure 4.50 from Molecular Parameters for Organosilicon Compounds Calculated from Ab Initio Computations, Grigoras S and T H Lame, Journal of Computational Chemistry 9 25-39, 1988. Reprinted by permission of John Wiley Sons, Inc. 

The coefficients A and jj, were introduced by Lame and named after him. They are related to elastic parameters of a medium, the Young s modulus E and the Poisson s ratio cr, by the formulae (Love, 1944 Udias, 1999)... 

Consider the following boundary value problem express the displacement vector field U in some domain V in terms of the values of U and of its normal derivative d J/dn on the inner side of the surface S bounding this domain. The constant elastic parameters of a homogeneous medium, Cp and Cj, are assumed to be known. The external volume forces F are distributed within some domain D, which is located inside V D C V), so the field U in Z satisfies the Lame equation... 

We can get a better understanding of the physical meaning of the elastic adjoint FV chet operator if we consider the first iteration in the inversion scheme (15.238). Let us assume that the initial distribution of Lame velocities in the model is given by some background parameters Cpt, (r) and Csb (r) ... 

In a last step the model is completed by an appropriate choice of the free energy function. The basic idea is to enhance the free energy of a Hnear elastic material [Eq. (18)] in such a way that the variation of the effective stiffness on the stractural parameter is obtained, i.e., we assume that the Lame constants depend on k in the form of Eq. (19). 

The various moduli and other parameters that can be formed out of Lame s constants in elastic theory may be generalized to the viscoelastic case by simply substituting fi (co), A (co) for //, A, and, if one wishes to return to the time representation, inverting the Fourier transform. An important example is v(o ), given by... 

Parameters A and are Lame coefficients related to Poisson s ratio v and Young s modulus E ... 

The two fundamental constants used in elasticity theory are the Lame constants A and jut, and the commonly used material parameters can be expressed in terms of these two constants. However, it is convenient to introduce into engineering practice a third materials constant, Poisson s ratio, V, which strictly is valid for simple stress fields where there are no shear components and only a single main tensile stress. The second Lame constant is then equivalent to the shear modulus G which then becomes related to the Young s modulus by the expression ... 

The linear isotropic approximation is widely used in calculations on the elastic behaviour of solids and the equations [2.14] are often expressed in terms of two material parameters known as the Lame elastic constants. 

Grid-based reservoir core properties total porosity, absolute permeability, formatitHi resistivity factor and corresponding cementation exponent m, elastic moduli assuming isotropy (bulk modulus k, shear modulus p. Young s modulus E, Poisson s ratio o, and Lame s parameter X), and corresponding acoustic velocities. 

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